Existence and concentration of positive harmonic functions with nonlinear boundary condition in expanding domains

Abstract

This thesis investigates the existence of a positive harmonic function uϵ defined in the rescaled domain Ωϵ = ϵ−1Ω, subject to a nonlinear boundary condition, where ϵ > 0, and Ω is a bounded domain in Rn, with n ≥ 3. In the case where ϵ → 0, corresponding to expanding domains, it is established that there exists a constant ϵ0 > 0 such that for all ϵ ∈ (0, ϵ0), the principal problem admits a nonconstant, positive least energy solution uϵ. Moreover, it is demonstrated that uϵ exhibits a power-type decay and an exponential decay in the first n − 1 variables. Furthermore, in the case where ϵ → 0, it is shown that the points where uϵ attains its maximum concentrate around a point of maximum for the mean curvature of ∂Ω. In contrast, in the case where ϵ → ∞, corresponding to contracting domains, it is proven that there exists a constant ϵ∗ > 0 such that for each ϵ > ϵ∗, the unique positive solution uϵ of the principal problem is constant. To conduct the analysis developed in Chapter 2, it is essential to investigate the existence of a ground state solution for the limit problem. Chapter 1 is dedicated to this study, addressing the problem of the existence and nonexistence of harmonic functions in the upper half-space, subject to an indefinite nonlinear boundary condition. The existence of a ground state solution is established, which is radially symmetric and exhibits exponential decay in the first n − 1 variables. Furthermore, in Chapter 1, an investigation is carried out on the existence and nonexistence of weak solutions in alternative cases that are not directly related to the problem considered in Chapter 2. In one such case, variational minimization techniques are employed to demonstrate the existence of a nontrivial weak solution. Additionally, a theorem is presented that characterizes the nonexistence of solutions under certain conditions.